An implicit wetting and drying approach for non-hydrostaticbaroclinic flows in high aspect ratio domains

Adam S. Candya,b,∗

aDepartment of Earth Science and Engineering, Imperial College London, UKbEnvironmental Fluid Mechanics Section, Faculty of Civil Engineering and Geosciences, Delft University of Technology, The Netherlands

Abstract

A new approach to modelling free surface flows is developed that enables, for the first time, 3D consistent non-hydrostaticbaroclinic physics that wets and dries in the large aspect ratio spatial domains that characterise geophysical systems. Thisis key in the integration of physical models to permit seamless simulation in a single consistent arbitrarily unstructuredmultiscale and multi-physics dynamical model. A high order continuum representation is achieved through a generalGalerkin finite element formulation that guarantees local and global mass conservation, and consistent tracer advection.A flexible spatial discretisation permits conforming domain bounds and a variable spatial resolution, whilst atypicaluse of fully implicit time integration ensures computational efficiency. Notably this brings the natural inclusion of non-hydrostatic baroclinic physics and a consideration of vertical inertia to flood modelling in the full 3D domain. Thishas application in improving modelling of inundation processes in geophysical domains, where dynamics proceeds overa large range of horizontal extents relative to vertical resolution, such as in the evolution of a tsunami, or in urbanenvironments containing complex geometric structures at a range of scales.

Keywords: Wetting and drying, Non-hydrostatic, Baroclinic, High aspect ratio domains, Multi-scale simulation,Vertical inertia, Finite element method

1. Introduction

Flooding has huge impacts on the economy of a regionand the livelihood of its people. Significant progress hasbeen made to model and predict the impact of these in-undation events, capturing the character of their sourceand resultant behaviour. Many challenges still exist andin particular in concurrently simulating the physical pro-cesses involved from the large planet-scale forcings downto the small human scales of an urban environment. Thisis highlighted in the review [1] as one of the key limita-tions of existing wetting and drying (WD) models. In anurban flooding scenario for example, modelled water col-umn depth could be down to 1cm over a horizontal rangeof tens or hundreds of kilometres, leading to a very highaspect ratio of ∼10−7.

Inundation flow models typically use simplified formula-tions of the Navier-Stokes equations, commonly the Saint-Venant shallow water equations (SWEs). These simplifi-cations make assumptions, such as a hydrostatic pressureand well-mixed water column, which are not necessarilyvalid in the whole range of scales relevant to the inun-dation. Non-hydrostatic processes become important, forexample, in the dispersive effects of short waves where the

∗Corresponding author address: Adam S. Candy, now at: En-vironmental Fluid Mechanics Section, Faculty of Civil Engineeringand Geosciences, Delft University of Technology, The Netherlands.

Email address: a.s.candy@tudelft.nl (Adam S. Candy)

ratio of vertical and horizontal scales of motion are notsufficiently small. The study of [2], considering the 2011Tohoku tsunami in Japan, found it was critical to includenon-hydrostatic effects to correctly model processes on thesmall scale, a point further highlighted by [3].

Multi-physics over a broad range of scales is typicallyapproached using multiple model runs at a hierarchy ofscales such that domains are nested, with varying com-plexity and included physics. As an alternative, efforts tointegrate the physics and scales of separate models into sin-gle Earth system models is growing, where it is importantindividual components function in a general context, andare not too restrictive in discretisation choice. Althoughthis can be achieved weakly with offline communicationbetween models, the ‘holy grail’ is a flexible single modelcapable of simulating a range of physics and scales, withinherit consistency and conservation.

This work pushes the boundaries in two key regards:firstly, adding a novel approach to WD in the ‘thin-film’family solving a full 3D pressure rather than the usualSWE approximation in very challenging acutely large as-pect ratio domains typical of geophysical systems — afirst for WD. Secondly, this brings the modelling of WDprocesses together with non-hydrostatic baroclinic flowdynamics in a single simultaneous and seamless systemmodel. This is critical for tightly coupled processes, forexample in tracking grounding line movement under anice shelf ocean cavity, that is strongly influenced by non-

Preprint submitted to Advances in Water Resources Monday 23rd January, 2017

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Figure 1: A schematic of an example high aspect ratio geophysical inundation domain considered here, with a ‘horizontal’ length scale Lspanning its extent on a geoid surface, and the ‘vertical’ length scale H. In reality, these length scales differ by many orders of magnitude.

hydrostatic and baroclinic ocean flows.Accurately tracking an inundation interface is techni-

cally challenging. Of the Eulerian type WD approaches[reviewed in 1], where the underlying spatial discretisa-tion is predominantly independent of space and time suchthat matrix operators can be cached and there is no needfor complex contour tracking, four types exist: elementremoval, limiting the computational domain to the wet re-gion (see [4], UnTRIM [5], [6, 7] and the WASH123D code[8]); thin film approaches (see [9], the FVCOM model [10],POM model [11] and also [12]); depth extrapolation fromwet to dry cells [13, 14]; and negative depth [15, 16] ap-plied in ROMS [17], including the use of porous mediabelow the sea bed [18, 19] and bathymetry movement [20].

Underlying model discretisations largely steer thischoice, with the first by far the most common for explicittime stepping models, applied in QUODDY, ADCIRC,MIKE21, Delft3D and in one of the first Eulerian methods[21], subsequently reviewed in [22]. Whilst robust, stabil-ity constraints restrict movement of the interface to onecell per time step (∆t), since the Courant number in dry-ing regions must be maintained less than one [23] to ensurea non-negative bound on water depth, a strict limitationon ∆t [24]. Depth extrapolation also suffers this restrictionwith elements switching states [1], whereas thin film andnegative depth options can be time-integrated implicitly.

For spatial discretisations, WD procedures were first ap-plied to structured meshes [4, 23], with updates to includenon-hydrostatic corrections [25], baroclinic solvers [17] andrecently subgrid information [6, 7, 26, 27] to include higherresolution bathymetry and flux calculations.

Current approaches to unstructured mesh geophysicalfluid modelling are considered in detail in [28], with theirpotential importance best highlighted in [29]. Indeed, thisreview states that whilst unstructured mesh models maynot replace structured modelling approaches completely,there are cases where this type of approach could be op-timal. In particular, allowing a flexible approach to the

vertical discretisation could improve accuracy and modelefficiency in domains where there are sharp changes inbathymetry relative to horizontal spatial resolution, strongnon-hydrostatic gradients in pressure, strong vertical iner-tial flows, or when it would be more optimal to reduceor increase the number of layers in shallow and deep re-gions, respectively. Moreover, these could be critical inthe fringes of the ocean boundary, along geometricallycomplex coastlines and in interactions with other typesof physical systems, such as an urban environment or thecomplex shallowing in ice shelf ocean cavities. Within thisdiscretisation type, WD models can more accurately modela wider range of scales in larger domains.

One of the early finite volume (FV) approachesUnTRIM [5] permits unstructured meshes with the con-straint that, like structured models, the domain elementsare orthogonal where circumcentres are inside their respec-tive elements. Its non-hydrostatic advance [30] is appliedin the SUNTANS model, with the same orthogonality re-striction. It contains a WD algorithm [31] stabilised witha technique from [19] that applies an increased drag tosatisfy an additional constraint on volume flues in dry re-gions. Similar approaches are also made in [32, FVCOM,MIKE21] with non-hydrostatic corrections added [e.g. 33].FV is low order only and models generally explicit.

Unstructured finite element (FE) methods offer high or-der continuum approximations which are more accurateand naturally include less diffusive and dispersive advec-tion schemes. WD has been built into 2D barotropicflow FE models such as QUODDY with dry element re-moval in [34]; ADCIRC, a SWE method for explicit hy-drostatic modelling of storm surges with dry removal [35];TELEMAC, initially using element removal [36] and nownegative depth; and SLIM [20] with a repositioned sea bedSWE method and adoption of implicit ∆t advance.

WD is combined with solvers capable of modelling baro-clinic processes in [17, 37], with the former using thin filmhigh order FE and the latter explicit finite difference with

2

negative depth WD and mode splitting. The former per-forms well in relatively modest aspect ratio domains butperformance is strictly limited by the following: use ofdirect solvers (LU decomposition), restrictions on dry el-ement aspect ratios and erroneous unphysical flows thatdevelop in dry regions.

Here a general approach for WD with FEs is consideredin full 3D, building on established methods for modellingfluid flow on fully 3D unstructured meshes [38] which varyin resolution and support a multiscale of physical pro-cesses, including non-hydrostatic and baroclinic dynam-ics in the large aspect ratio domains found in geophysi-cal domains. Under the constraints of a global numberof degrees of freedom, this allows the focus of computa-tional resources on small scale regions and areas of inter-est, whilst capturing the large scale flows elsewhere in thedomain. An additional advantage is that there is neithera constraint on the internal mesh structure, nor is it fixedin time. It is not constrained to layers, and can be com-pletely (or partially in select regions) fully anisotropicallyunstructured.

To allow efficient time integration over a range of el-ement sizes, an implicit treatment necessitates a contin-uum approach to interface tracking. A thin film is ap-plied, which as [1] notes, generally satisfies mass andmomentum conservation without significant special treat-ment, and produces a realistic and smooth wetting front.WD is included in a natural manner, through additionalterms in the momentum equation and modified bound-ary conditions. Indeed, the numerical treatment is care-ful to ensure the solution remains in the Sobolev solutionspace of the original physically-based weakly formulatedGalerkin problem. Prognostic variables, including trac-ers, are self-consistent through the FE formulation andnotably, through use of a combined pressure variable, con-sistency with the free surface is naturally inherent.

In the following sections 2 to 4, the new consistent ap-proach for WD in large aspect ratio geophysical domains isdeveloped, with details of mesh movement in section 5 andadditional strategies noted in section 6. This is validatedin section 7 and evaluated in section 8.

2. Governing continuum equations

2.1. 3D Boussinesq with piezometric pressure

The non-hydrostatic Boussinesq equations for a rotat-ing stratified fluid, are solved in a time-dependent domainΩ ⊂ R3 (see figure 1), bounded by the surface Γ. This issplit into the free surface boundary Γη, and the remainingbound Γb = Γ \ Γη. These are defined for the prognosticvariables of velocity u : Ω × [0, T ) 7→ R3, and pressurep : Ω× [0, T ) 7→ R, over the time interval [0, T), such that

ρ0

(∂u

∂t+ u · ∇u

)−∇ · µ∇u+∇p = −gρ′ng, (1)

∇ · u = 0, (2)

where µ is the tensorial dynamic viscosity, −ng and g thegravitational acceleration direction and magnitude respec-

tively, and ρ : Ω × [0, T ) 7→ R the density. The latter issplit into a background ρ0, and perturbation density ρ′,such that ρ = ρ0 + ρ′. Since the hydrostatic component ofpressure of the equilibrium state does not have an impor-tant contribution dynamically, it is subtracted from themomentum equation and the full pressure p, is replacedby a piezometric pressure, commonly applied in coastalengineering applications [e.g. 39], defined as

p := p+ ρ0gng · r + pa, (3)

for a position vector r, relative to a position where hy-drostatic pressure is zero. Atmospheric pressure at theinterface is denoted pa.

Redefining the prognostic pressure with this particu-lar choice of piezometric pressure forms a combined freesurface – pressure prognostic p(p, η) eliminating the needto solve a separate, commonly used, wave equation forthe free surface, denoted by the injective function η :Ω × [0, T ) 7→ Γη. The prognostic pressure p now containsnon-hydrostatic components and the hydrostatic pressuredue to perturbations in the free surface elevation. Thisremaining hydrostatic pressure ρ0gη, is the boundary con-dition for p at Γη, and through (3), we find

p∣∣Γη

= ρ0gη. (4)

2.2. Boundary conditions

With the inclusion of the free surface height in the prog-nostic pressure, the kinematic free surface boundary con-dition of appendix A is expressed

n · ng∂p

∂t

∣∣∣∣Γη

= ρ0gn · u, on Γη. (5)

This is the boundary condition for η and now a requiredconstraint for the combined p(p, η) prognostic variable.This is joined by the u and p boundary constraints

u · n = 0, ∀x ∈ Γb, and

p = pa, ∀x ∈ Γη. (6)

More general conditions, for open ocean boundaries or fluxinputs, can be applied without fundamental changes to theapproach.

2.3. Coordinate system and frame of reference

Note additionally that the direction of gravity, describ-ing the normal ng, is not restricted to a Cartesian z-component, such that the development is relatively inde-pendent of the coordinate reference frame. It is free tovary arbitrarily within R3, aligned with the local directionof gravitational acceleration, and it is possible for example,to apply this method to the spheroid shell of the Earth ina Cartesian coordinate reference frame.

3. Spatial and temporal discretisation

The non-linear system of equations (1) and (2), com-bined with boundary conditions (5) and (6), are solved forp(p, η), and velocity u, using a Chorin projection method[40] to enforce incompressibility. This is a modified predic-tor – corrector scheme based on [41] in which a predictor

3

un+1∗ is obtained from momentum conservation that is not

divergence free, such that a correction un+1 = un+1∗ −∇φ

is then calculated subject to the divergence-free constraint∇·un+1 = 0. For each time step, this proceeds for a num-ber of Picard iterations until sufficiently converged.

3.1. Temporal discretisation

Discretisation in time is achieved by the θ-method [42] inall cases, for a time step ∆t, such that the explicit forwardEuler, Crank-Nicolson and backward Euler time-steppingschemes can be obtained with choices of θ = 0, 1

2 and1, respectively. The modified Navier-Stokes with implicitfree surface system (1) and (2) at a time n is therefore

ρ0un+1 − un

∆t= Rn+θ −∇pn+θ − ρn+θgng, (7)

∇ · un+1 = 0, (8)

where Rn+θ = θRn+1 + (1 − θ)Rn contains the advectivemass flux term, together with viscosity and other sourceterms. A choice θ ∈ [ 1

2 , 1] leads to an implicit time step-ping scheme that allows simulations to use large time steps,which are not restricted by the Courant-Friedrichs-Lewy(CFL) condition [43] with respect to the velocity and wavespeed. In practice for the simulations presented here, forthe required level of accuracy and stability, Courant num-bers up to 10 are applied.

3.2. Combined free surface – pressure Chorin corrector

Under a Galerkin FE spatial discretisation the tempo-rally discretised momentum (7) and continuity (8) equa-tions are tested with the velocity φi and pressure ψi basisfunctions, respectively. The trial functions u and p aredefined in terms of their respective basis functions also,and appendix B describes their form and the nomencla-ture used in more detail. This leads to the space-timediscrete momentum equation

ρ0Mu

∆t

(un+1∗ − un

)+ θAn+1un+1

∗ + (1− θ)Anun

= θCpn+1∗ + (1− θ)Cpn + Su, (9)

ρ0Mu

∆t

(un+1 − un

)+ θAn+1un+1 + (1− θ)Anun

= θCpn+1 + (1− θ)Cpn + Su, (10)

for a Picard iteration step and end of time step, respec-tively. The starred variable un+1

∗ is the current best ap-proximation to un+1, calculated from pressure at the pre-vious time level n. The best guess of the solenoidal veloc-ity at a time level n+1 is denoted un+1, and used in thecalculation of updated non-linear operators, such as massflux An+1. The velocity space mass matrix Mu addition-ally contains the diagonal or block-diagonal (depending onthe chosen discretisation) component of viscosity from R,which is to be treated implicitly in pressure. The discretecross-space gradient operator Cij := −

∫Ωφi∇ψj dΩ, con-

tains an inner product over velocity and pressure spaces,leaving sources in Su.

Subtracting (9) from (10) and multiplying by θ∆tCTM−1u

yields a discrete Poisson equation for the correction

ρ0θCT(un+1−un+1

∗ ) = θ2∆t CTM−1u C(pn+1−pn+1

∗ ). (11)

3.3. Discrete continuity

Discretisation of the continuity equation (8) is written

θCTun+1 + (1− θ)CTun +GTθ un+1 +GT1−θu

n = 0, (12)

where Gθ,ij :=∫

Γηθnφiψj dΓ and G(1−θ),ij :=

∫Γη

(1 −θ)nφiψj dΓ. For Gθ = G(1−θ) = 0, the system of equa-tions enforces incompressibility with weakly applied nonormal flow boundary conditions.

3.4. Discrete modified kinematic boundary condition

Discretisation of the free surface boundary condition (5)is now required to provide the boundary integral terms in(12), with a θ time discretisation described by

nn+1 · ngpn+1 − nn · ngpn

= ρ0g∆t(θnn+1 · un+1 + (1− θ)nn · un

). (13)

Discretisation of (13) in space using the test and trial func-tions φi and ψi, introduced in section 3.2 gives

Mspn+1 − pn

ρ0g∆t= GTθ u

n+1 +GT1−θun, (14)

with the surface integral Ms,ij :=∫

Γηngψiψj dΓ.

Applying this discrete combined p(p, η) kinematic con-dition (14) to the discrete continuity (12), we find

θCTun+1 + (1− θ)CTun +Mspn+1 − pn

ρ0g∆t= 0. (15)

Substituting (15) into the momentum equation (11), withthe correction defined ∆p := pn+1 − pn+1

∗ , yields(θ2CTM−1

u C +Ms

g(∆t)2

)∆p

= −θCTun+1∗ +(1−θ)CTun

∆t− Ms

g(∆t)2(pn+1∗ − pn). (16)

3.5. Combined free surface – pressure system solution

During a single Picard iteration, the first velocity predic-tor step solves the discrete linearised momentum equation(9), to establish an updated intermediate velocity un+1

∗ ,from the current best approximation to the velocity andpressure, and their value at the previous time step.

The predictor un+1∗ obtained is not divergence free in

general, and in order to enforce the incompressibility con-dition, a pressure correction ∆p is calculated to projectthis velocity into the divergence free subspace by solving(16) above. The velocity correction is made to update theintermediate velocity, consistent with the new intermedi-ate pressure, and projected to the divergence free subspaceusing the difference of (9) and (10), where

un+1 = un+1∗ +

θ∆t

ρ0M−1u C∆p. (17)

Finally, the interface tracking step adjusts the free sur-face position following (4) in light of the new pressure field,in a direction −ng, parallel to the gravitational vector.

4

4. High aspect ratio wetting and drying domains

4.1. Wetting and drying of the simulation domain

The free surface boundary is split into distinct wet anddry regions (illustrated in figure 1), defined by the com-bined p(p, η) at the surface such that Γη = Γw ∪ Γd, with

Γw : r ∈ Γη, ∀ p(r) ≥ ρ0g(h(r) + d0), and

Γd : r ∈ Γη, ∀ p(r) < ρ0g(h(r) + d0).

The conditions in dry regions differ from those in wetin two defining ways. Firstly, the water column depthis maintained at a threshold minimum d0 above the bot-tom bathymetry defined by h : Ω 7→ R, and secondly, thesurface boundary condition on the combined p(p, η) prog-nostic variable is modified to enforce this constraint in thesolver. With the depth constraint the free surface evolu-tion described by (4) provides the first constraint

η(r) = max

(1

ρ0gp(r), h(r) + d0

), for r on Γη. (18)

The second is found by modifying the combined kinematiccondition (5), which in light of the depth restriction gives

n · ng∂

∂tmax (p, ρ0g (h+ d0)) = ρ0gn · u, on Γη. (19)

In wet regions Γw, the constraints (18) and (19) reduceback to the free surface conditions (4) and (5), respectively.In dry regions Γw, (19) is a no normal flow condition, andeffectively imposes a rigid lid approximation.

4.2. Conditioning of the pressure calculation

The spatial domains of geophysical processes are typi-cally large aspect ratio, due to the gravitational influenceand disparity in dynamics parallel and perpendicular togeoid surfaces. The solution of a non-linear fluid flow sys-tem in these types of domains including non-hydrostaticdynamics, with implicit time evolution and a predictor –corrector approach such as section 3.5 is shown in [44] tolead to an ill-conditioned pressure system. In the limit oflarge domain aspect ratio and long time steps, the systembehaves approximately as though it has a rigid lid, wherethe free surface is fixed with η = 0 and u · n = 0.

The dry regions introduced by the WD process signifi-cantly exacerbate ill-conditioning, since a rigid lid condi-tion is applied directly and the region contains elementswith acutely large aspect ratios due to their defining shal-low water column depth.

The correction un+1 = u∗ −∇φ (17) is calculated sub-ject to the divergence-free constraint ∇ · un+1 = 0. Thisleads to the following pressure Poisson equation for φ

∇2φ =∇ · u∗, (20)

which corresponds to the discrete Poisson operatorCTM−1

u C in the formulations (16) and (42) above. Forno normal flow boundary conditions where u · n = 0, atinterfaces with bedrock or in the case of the rigid lid ap-proximation for the ocean-air interface, the coupling be-tween velocity and pressure results in the corresponding

boundary condition on (20) as the Neumann expression

∂φ

∂n= 0, on Γη, (21)

ensuring the velocity constraint is consistently preserved.Applying a kinematic condition instead leads to the ho-

mogeneous Dirichlet condition φ = 0 on Γη. The redefi-nition of pressure in (3) to form the piezometric pressurehere allows standard pressure splitting approaches to treatbaroclinic and barotropic dynamics [e.g. 45] in the gen-eral case of domains discretised with fully-unstructuredmeshes. These schemes themselves aid the conditioning ofpressure solves in geophysical models [46], where there isa large disparity of scales and resolution of the dominantphysical processes. This piezometric variable satisfies thesame equation (20), with a modified right-hand side sourceterm and boundary condition.

Discretisation of the kinematic condition (5) defined interms of the piezometric pressure using implicit backwardEuler in time gives a Robin condition for φ, such that

n · ngφ

∆t2= g

∂φ

∂n. (22)

With the barotropic wave speed c =√gH, for a distance

H, and noting that

φ

∆t2

/g∂φ

∂n≈ H

g∆t2=

(H

c∆t

)2

,

the ratio of the terms in (22) scale as the square of thetime it takes for a barotropic wave to travel a distance Hrelative to the length of a time step, and we see that thecondition for free surface flows (22) tends to that of therigid lid (21) in the large time step limit. So although ad-justing a system to apply a free surface kinematic bound-ary condition on the top surface as opposed to a rigid liddoes improve conditioning for modest aspect ratios, as thedisparity in scales increases and the aspect ratio becomessmaller, or equivalently larger time steps are taken, theill-conditioning of a rigid lid system is soon recovered dueto the quadratic dependence.

The multigrid preconditioner of [44] for unstructuredmeshes on high aspect ratio domains helps better conditionthe Poisson problem in general, without consideration ofWD, using a combination of algebraic multigrid and a geo-metric vertical prolongation operator. This solver methoditself makes it feasible to run non-hydrostatic unstructuredmesh simulations of fluids in geophysical domains.

Whilst the relatively moderate aspect ratio wet areascan be treated by the multigrid preconditioner approach,specific methods to handle the acute aspect ratio and di-rect rigid lid condition applied in dry regions are required,if this general fully 3D and non-hydrostatic WD approachis to be applied to real geophysical systems.

4.3. Quantification of the ill-conditioning

The discrete form of the Laplacian operator that ap-pears on the left-hand side of (20), seen in (16), has eigen-values λi ∼ k2

i , for wavenumbers ki. The conditioning

5

of the matrix is determined by the ratio of the maximum‖λ‖∞ and minimum ‖λ‖min eigenvalues. This is, equiva-lently, the ratio of the minimum and maximum wavenum-bers, kmin and kmax, squared

κ(CTM−1

u C)

=

∣∣∣∣ ‖λ‖∞‖λ‖min

∣∣∣∣ ∼∣∣∣∣∣∥∥k2∥∥∞

‖k2‖min

∣∣∣∣∣ =

∣∣∣∣ ‖k‖∞‖k‖min

∣∣∣∣2. (23)

For high aspect ratio problems H/L 1, for H and Lcharacteristic length scales of the solution domain in thevertical and horizontal, respectively (see figure 1), we find

‖k‖min ∼1

H, and ‖k‖∞ ∼

1

L, and hence

κ(CTM−1

u C)∼(H

L

)2

. (24)

On a spheroid, such as the Earth, the characteristic ‘hor-izontal’ length scale L is the extent of the encompassingsurface geoid, with H the height in a direction parallel togravitational acceleration. Conditioning of linear systemthat results from the discretisation of the Poisson equationis approximately proportional to the square of the aspectratio of the global domain. Equivalently, the element edge-lengths, which are constrained to resolve processes impor-tant to the simulation, can also be used to characterisethe scaling, such that condition number is proportional to(∆x/∆z)2, with ∆x and ∆z characteristic edge-lengths inlocal horizontal and vertical directions, respectively.

Entries into the matrix of the linear system that arisefrom dry cells would ideally be removed, in a process simi-lar to lifted Dirichlet boundary conditions [e.g. 47] and thesolver limited to variables on the wet sub-system, much likean element removal approach. For an implicit approachit is not clear how this would be accomplished withoutadversely affecting the natural evolution of the interface.Instead, under implicit integration, treatment of the ill-conditioning highlighted by (24) needs to be addressed.

4.4. Vertical velocity relaxation in dry areas

To close the system (1)–(2), an equation of state is re-quired. Details of the form of this function do not influencethe development that follows, and a general treatment ofthe evolution of density is considered, such that

∂ρ

∂t+ u · ∇ρ = 0, (25)

with its temporal discretisation following section 3.1 as

ρn+1 − ρn

∆t+ un+1 · ∇ρn+1 = 0. (26)

Development of the approach proceeds with a discretisa-tion of the density transport equation (25), in a slightlydifferent linearisation to that of (26), of the form

ρn+1 − ρn

∆t+ wn+1 ∂ρ

n+1∗∂z

+ sn+1ρ∗ = 0, (27)

ρn+1∗ − ρn

∆t+ wn+1

∗∂ρn+1∗∂z

+ sn+1ρ∗ = 0, (28)

with vertical velocity w, starred variables representing thebest current guess, and the source term sn+1

ρ containingdetails of spatial gradients of density locally aligned tothe geoid. Subtracting (28) from (27) gives a transportequation that mirrors (11), describing the variation overthe Picard iteration process

ρn+1 − ρn+1∗

∆t+ (wn+1 − wn+1

∗ )∂ρn+1∗∂z

= 0. (29)

Substitution of this temporally discrete density transportequation (29) into the momentum equation (7) leads to

ρ0un+1 − un

∆t= Rn+θ −∇pn+θ

+ gng∆t∂ρn+1∗∂z

(wn+1 − wn+1∗ )− ρn+1

∗ gng. (30)

The FE weak form of (30) is developed by testing withvelocity basis functions φi and applying integration byparts twice at the free surface to obtain∫

Ω

φiρ0un+1 − un

∆tdΩ =

∫Ω

φi

(Rn+θ −∇pn+θ

+ gng∆t∂ρn+1∗∂z

(wn+1 − wn+1∗ )− ρn+1

∗ gng

)dΩ

−∫

Γη

φin · ngg∆t(ρn+1 − ρa)(wn+1 − wn+1∗ ) dΓ. (31)

The density of air just above the free surface interface ρa,can in most cases be neglected as a small effect, in thesame way as the atmospheric pressure.

Assuming that, for shallow waters, the vertical velocityw is linearly related to the distance from the bottom of theocean or in a depth-averaged sense and ignoring the den-sity variations ρ′ in the surface integral above, the termsin (31) above containing explicit reference to the verticalvelocity w can be grouped into an absorption term

σzz = g∆t max

(0,−∂ρ

n+1∗∂z

)+g∆tρ0

dn · ng, (32)

where d = h+ η is the water depth, such that∫Ω

φiρ0un+1 − un

∆tdΩ =

∫Ω

φi

(Rn+θ −∇pn+θ

− σ(un+1 − un+1∗ )︸ ︷︷ ︸

†

−ρn+1∗ gng

)dΩ,

with σ =

0 0 00 0 00 0 σzz

. (33)

The inverse time scale for the vertical velocity relaxationis defined by (32). As the Picard iterations proceed andun+1∗ → un+1, the magnitude of this stabilising term,

marked by † in (33), relaxes to zero. Although the absorp-tion coefficient σ will be relatively small in wet regions,and the contribution from † small overall, it is importantto include these terms throughout in order to maintainconsistency and as a result, accuracy.

The following conditions on vertical density gradient,the free surface, vertical viscosity, and vertical absorption

6

provide a well-conditioned pressure Poisson equation

1. Vertical density gradient

(∆x)2

(∆z)2≤ a2g∆t2max

(∂ρn+1∗∂z

, 0

), (34)

2. Free surface variation

(∆x)2

(∆z)2≤ a2(∆t)2g

d, (35)

3. Vertical viscosity

(∆x)2

(∆z)2≤ a2∆t νzz

∆z2, (36)

4. Vertical absorption

(∆x)2

(∆z)2≤ a2∆t σzz, (37)

where a is a tolerable aspect ratio of element length scales(e.g. unity in the isotropic case), ∆x and ∆z characteriselocal resolution scales, νzz is a kinematic viscosity, andσzz an absorption. Note that the viscosity of (36) mustbe treated implicitly or semi-implicitly in pressure (e.g.diagonal or block diagonal) in order to control the condi-tion number of the pressure Laplacian. Implementation ofthe viscosity in stress form is appropriate here since tensorforms directly smooth horizontal velocities in the vertical.

In the case of WD, where (35) does not hold, we must en-sure (37) is satisfied by a suitable choice of the absorptionσzz. From (37), in order to make the resulting pressurematrix feel like an O(a) aspect ratio domain, we need

σzz =(∆x)2

a2∆t(∆z)2. (38)

The form of σzz in (38) defines the inverse time scale forthe vertical velocity relaxation in (32). Note that this formof σzz is spatially varying, and in particular the character-istic local length scales ∆x and ∆z are non-homogeneousacross the geoid surface. In WD simulations these fieldscontain large deviations which are indicative of the regionsaffecting conditioning of the pressure Poisson equation.

4.5. Discretisation for high aspect ratio domains

The momentum equation (33) discretised in space-timeat any given Picard iteration step is

ρ0Mu

∆t

(un+1∗ − un

)+ θAn+1un+1

∗ + (1− θ)Anun

= θCpn+1∗ + (1− θ)Cpn −Q

(un+1− un+1

∗)

+ Su, (39)

with Qij :=∫

Ωφiσφj dΓ. This balance compared to its end

of time step state is multiplied by θ∆tCTM−1u , to give(

ρ0θCT − θ∆t CTM−1

u Q) (un+1 − un+1

∗)

= θ2∆t CTM−1u C(pn+1 − pn+1

∗ ). (40)

This is equivalent to (11) previously, noting that the termmarked † in (33) is zero at the end of a time step.

The new form of the combined kinematic boundary con-dition (19) leads to a time discretised form modified from

(13) to include the no normal flow component applied indry regions and is described by

nn+1 · ngmax(pn+1, ρ0g (h+ d0)

)− nn · ngmax (pn, ρ0g (h+ d0))

= ρ0g∆t(θnn+1 · un+1 + (1− θ)nn · un

).

Moreover, the surface integral Ms is modified such thatthe discrete modified kinematic condition (14) becomes

Mwpn+1 − pn

ρ0g∆t+Md

h+ d0

∆t= GTθ u

n+1 +GT1−θun,

where Mw,ij =∫

Γwngψiψj dΓ and Md,ij =

∫Γdngψiψj dΓ.

This kinematic condition change modifies the pressure cor-rection and the discrete continuity (15) becomes

θCTun+1+(1−θ)CTun+Mwpn+1−pn

ρ0g∆t+Md

h+d0

∆t= 0. (41)

Substituting (41) into momentum (40) yields the discretecombined p(p, η) Poisson corrector, with (16) evolving to(

θ2CTM−1u C +

Mw

g(∆t)2

)∆p

= −θCTun+1∗ + (1− θ)CTun

∆t−Q(un+1∗ − un+1

)∆t

− Mw

g(∆t)2(pn+1∗ − pn)− ρ0Md

h+ d0

(∆t)2. (42)

The predictor – corrector method of section 3.5 solves thenon-linear system with the updated p(p, η) Poisson cor-rector (42) combined with discrete linearised momentum(39), and a velocity correction determined from (40).

4.6. Self-consistency and physical basis of the solutionMass, momentum and tracer quantities are self-

consistent and conserved, properties inherited from theirunderlying Galerkin FE weak formulations [38] and useof a thin-film [1, 37]. The constrained discrete Sobolevsolution space of the weak form modified with the addi-tional terms marked † in (33) converges on the solutionspace of the original form without these, as the the Picardprocess proceeds. In a similar manner to Petrov-Galerkinand variational multiscale [48] residual-based stabilisationmethods [49], this ensures consistency, that the solutionfound is a valid solution of the original weak Galerkin for-mulation, a true discrete solution to the governing contin-uum equations and is therefore physically-based.

Moreover, just like streamline-upwind Petrov-Galerkin(SUPG) stabilisation, the additional terms themselves aredefined from physical properties of the flow. For example,(32) includes contributions from g, the local vertical den-sity gradient and water column depth. This is supportedalong with local discretisation parameters such as timestep and element size used to quantify unresolved scales,in a similar way to multiscale turbulence closures [49].

4.7. Determination of characteristic length scalesAccurate calculation of the characteristic length scales

is critical to the success of the approach, particularly dueto the quadratic dependence in (38).

7

The calculation of the characteristic horizontal lengthscale could be simply the minimum or maximum edgelength of the element projected to a 2D horizontal geoid.A more accurate approximation can be determined fromthe smallest and largest circumscribing circular bounds ofthis projection. The length scale ∆x is then a function ofthese minimum and maximum extents. This is a naturalapproach for models employing anisotropic mesh elements.

The vertical length scale is less ambiguous to determine,since unique intersections with Γη and Γb exist ∀x ∈ Ω,due to the construction of geophysical domains [50, 51],and similarly for internal layers. Evaluating length scalefunctions at Gaussian quadrature points rather than byelement further increases accuracy, since FE assembly in-tegrations are performed this way, with options to developmean or area-weighted means. This is trivially extended tosuperparametric elements which are typically used in thetop layer for accurate representation of geoid curvature.

Arguably the best characterisation of tetrahedral ele-ment size is determined from the Jacobian transformationmatrix which projects a FE from global to local param-eterised space. The determinant of the transformationJacobian intersected with the local (to quadrature point)surface geoid plane and gravitational acceleration vectorwill give characteristic length scales for the element in therequired horizontal and vertical directions, respectively.This approach also naturally handles element anisotrophyand meshes which are fully unstructured in 3D. The meritsof this choice are examined in section 7.3.

4.8. Correction to velocity relaxation in shallow regions

Under no forcing the momentum equation (33) tends torelax the implicit velocity un+1

∗ to the state in the previoustime step un, but this can be too strong in very shallowareas. This is corrected by reducing the magnitude of theexplicit part of the velocity that we relax to, by adding−γun to the right of the momentum equation, with

γ = max

(2

(1− d

2d0

), 0

). (43)

For a water column depth d, this relaxation scales awaythe velocity in the vicinity of dry regions where d < 2d0,and relaxes to zero in dry regions, where d = d0.

5. Mesh movement with wetting and drying

5.1. Discrete function space updates

The free surface evolution results in many quantitiesvarying in time, such as the free surface normal vector nin (13). Moreover, this includes the mesh, and hence spa-tial discretisation, which leads to a change of the discretefunction spaces Sh, and their spanning basis sets resultingin new forms of mass and other matrices in discrete formssuch as (14). For conservation and accuracy it is neces-sary to update the discrete non-linear system during thePicard iteration to reflect these changes. There are varioustechniques to handle this conservatively, through the defi-nition of a grid velocity, for example. In this formulation,the domain discretisation is updated at the end of a Picard

iteration to reflect the new free surface height predicted,with the normal n, mass matrix and other matrices repre-senting advection and surface integrals recalculated underthe new domain discretisation. It is therefore the case thatthe discrete matrices Mu, Ms, C, A, G, and Q; free sur-face normal n, basis functions φ and ψ, domain Ω and freesurface Γη, are always the best known approximation, i.e.the starred n+1 case. A subtle exception is at the end ofthe final Picard iteration, where the update is not made,to ensure the domain and derivative parameters are thosethe prognostic variables were calculated on.

The generalised approach that includes the non-linearadvection term in the governing equations (1) precludesthe discretised spatial operator from being self-adjoint.Evaluation of this non-linear term requires sub-cycling,and for under-resolved high Froude number or rapidly-varying flows this could require a large number of iter-ations to converge, unless the continuum system is lin-earised, or local resolution increased.

5.2. Surface representation and interface tracking

At the end of each Picard iteration, as outlined in sec-tion 3, the free surface position is updated using (18) toreflect the new pressure at the interface pn+1

∣∣ηs

. Due to

the minimum threshold d0, the perturbation of the inter-face in the direction of the gravitational acceleration islimited. If the pressure pn+1 at the interface implies itshould move below this level, it is fixed at the thresh-old level above the bottom bathymetry (i.e. η = h + d0).The pressure remains unaffected, and is allowed to deviatefrom the interface position η. Conversely, as soon as pn+1

produces a water column depth greater than d0 the freesurface interface moves upwards. Correspondingly, the do-main discretisation is updated with the mesh stretched inthe direction −ng, parallel to the gravitational vector, tomeet the new free surface bound.

Spatial representation of η is inherited from the func-tion space used to approximate the combined p(p, η). Ir-respective of the order of variation, such as quadratic forthe PDG

1 − P2 element pair, the interface is approximatedby a piecewise linear function as far as the domain repre-sentation is concerned. This satisfies the min-max prop-erty, such that the extent of the surface is bounded bythe nodal positions that define its representation. This,together with the minimum threshold level prevents ele-ments from becoming inverted or excessively small.

5.3. Remeshing

It is not a requirement that the domain is remeshedanew to these adjusted bounds. Since only one of the do-main boundaries is perturbed through the above processand in a direction aligned to the gravitational vector field,locally orthogonal to other bounds of the domain, it ispossible to apply a relatively simple r-adaptive transform.The domain mesh is stretched linearly in this direction tofit the new boundary. It is also possible to limit the pertur-bation to the nodes on the free surface, or to apply more

8

complicated r- or h-adaptive strategies to achieve a hy-bridised coordinate system [see 52–54] for more accuratesolutions or better-represented features. The implemen-tation of the approach described here in the model code[Fluidity, 38] functions with and supports these methods.

6. Additional stabilising approaches for dry areasin high aspect ratio domains

Two supplementary approaches to control condition-ing are presented, acting directly to prevent strong erro-neous flows developing in the thin film and modifying be-haviour in neighbouring wet regions. This is exacerbatedby the fact the physical system is solved in a weak sense,which whilst better for conditioning can permit large fluxesacross the interface. Unlike the above, these approachestransform the solution space and have the potential to af-fect the solution in unphysical ways. They are presentedas additional techniques which can be employed to enablea solution to be reached, but require careful application.

6.1. Manning-Strickler drag and dry region stability

In the case of inundation flows where WD is applied, aparameterisation of drag that is commonly employed is theManning-Strickler formulation, defining the bottom stress

n · µ∇u = n2g|u|ud1/3

, on Γb, (44)

where n is the Manning coefficient, d is the water depthand n here is the unit surface normal on the bottom sur-face Γb. This formulation itself has a stabilising effect, andmore so in the very shallow dry regions, with a drag ap-plied along the bottom boundary proportional to d−1/3. Inpractice, the Manning-Strickler bottom stress is sufficientto prevent significant erroneous flow developing in dry ar-eas. In the cases of acute high aspect ratio, long time stepsor particularly steep bathymetric gradients, the Manningcoefficient can be increased in dry regions and their prox-imity to increase the stabilising effect, with

n = n+ max

(0, ndry

2d0 − dd0

),

where n replaces n in (44), and for ndry a new Manningcoefficient (with usual standard units of sm−1/3) large insize, relative to the standard coefficient n.

6.2. Horizontal bulk eddy viscosity in dry regions

A second solution to increase stability, is to damp flowdirectly in dry regions with a bulk volume viscosity or asource-absorption sponge, both allowing the approach toremain implicit.

This stabilisation is applied throughout the domain, orselectively in dry regions and their immediate proximity,with the large horizontal viscosity

νL = max

(0, νdry

2d0 − dd0

), (45)

introduced to control spurious horizontal fluxes, with νdry

a constant eddy viscosity coefficient and d ≥ d0 ∀x ∈ Ω.This horizontal viscosity is continuous in space without

discontinuous jumps in intensity across the WD interface,acting in the proximity of dry regions where d < 2d0.

7. Validation and application: Numerical testsPerformance of the implicit WD formulation described

in sections 2 to 5, and additional strategies of section 6are examined in four test scenarios in acutely high aspectratio domains, to a degree found in geophysical systems.

7.1. Implementation and verificationThe approach has been implemented and validated in

the FE fluid dynamics code Fluidity [38]. This simula-tion framework contains many tools for geophysical mod-elling, is parallelised with sophisticated load balancing andsupports adaptive mesh methods allowing computationaleffort to be focused on regions of dynamic interest. Itfunctions for a spatially variable gravitational accelerationvector, and hence can be used for large-scale simulationson the Earth’s spheroid. The implementation includes asuite of test cases to routinely verify the new algorithmin a formal sense, in an automated continuous verificationbuild engine [55] to ensure robustness of the code and re-siliency in light of further development. The unstructuredmeshes used in the following cases were built by means ofthe open source software Gmsh1.

The balance and LBB stability properties of the PDG1 −

P2 velocity-pressure pairing [see 56] aid conditioning andare used in all applications considered here. All four caseshave been run on the purely continuous pairing P1 − P1

also, but due to the pressure filtering required, did not per-form as well, and in all but modest aspect ratio cases weretoo ill-conditioned to reach convergence. The behaviour ofPDG

1 − P2 and P1 − P1 and their relative performance inregular aspect ratio problems is presented in [56].

Due to the aspect ratios considered, all cases usethe multigrid preconditioner described in [44] for itera-tive solution of the conditioned symmetric pressure Pois-son linear system in combination with Conjugate Gra-dient [CG, 57]. The momentum system is solved in amore standard approach with Symmetric Successive Over-Relaxation [SSOR, 58] preconditioning and the iterativeRestarted Generalised Minimal Residual [GMRES, 59] al-gorithm, where the calculation is restarted after k = 30iterations. The iterative SSOR-GMRES process is per-formed using algorithms built into the established andwell-verified PETSc library [60]. In contrast to the study[37], it was found that two Picard iterations provide suffi-cient convergence of the coupled system in the cases stud-ied. In all cases, both linear systems are solved to aconvergence criteria specified by a relative error toleranceof 10−7, which is considered sufficiently accurate. Thequadrature based subgrid resolution described in [37] isalso used, with a quadrature degree of eight.

7.2. First Balzano sloped channel benchmarkThe first two sets of numerical tests are from the suite of

problems in Balzano [22], selected since they exhibit the

1http://www.geuz.org/gmsh.

9

1.0

0.5

0.0

-1.0

-1.5

-0.5

-2.01.40.40.0 0.8 1.00.6 1.20.2

1.0

0.5

0.0

-1.0

-1.5

-0.5

-2.01.40.40.0 0.8 1.00.6 1.20.2

Figure 2: The first Balzanochannel flow benchmark with,in this presented case, ahorizontal extent of 1.38 ×104m, corresponding to a min-imum element aspect ratio of∼ 10−6. The discretised hor-izontal surface (a) contains108 triangular elements witha characteristic length scaleof ∼ 500m. Vertical sectionsshow the free surface positionat 10min intervals for the ini-tial drying phase (b) and dur-ing wetting (c).

Figure 3: Impact of the optimal aspect ratio parameter on solverconditioning in the first Balzano benchmark over a WD phase. 101individual simulations shown.

Figure 4: Pressure solver iterations to convergence in the firstBalzano benchmark with respect to domain global aspect ratio, in101 individual simulations, with and without conditioning applied.

problematic ill-conditioning in as simple a setup as pos-sible. No analytical solution is available, so the problemconfiguration is chosen consistently with [22] to be able todraw comparisons. The base benchmark case is developedfrom the originally 2D domain consisting of a 13.8km longslope with a depth of 5m at one end which tends to zero at

the other. Recently developed schemes, such as the flux-limiting WD method for FE SWE models presented in [61]and the non-hydrostatic algorithm proposed in [37], havebeen benchmarked on these cases. These model in 3D,but force dynamics to occur predominantly in the direc-tions where the extremes in extent occur, with 10 elementsintroduced in the third direction in the former and 1-2 inthe latter, which is followed here to a width of 1km. Withthe assumption solutions are laminar, this extrusion into3D space will not change the physical behaviour. Thesloped bottom bathymetry is defined h(x) = x/2760, forthe x-coordinate direction indicated alongside the surfacegeoid computational mesh in figure 2(a). The base casesingle-layer mesh contains vertically-aligned nodes and ahorizontal element size of 500m.

Following the benchmark description in Balzano [22](also in [61]), no normal flow boundary conditions areapplied at the bottom and shallow end of the domain,and additionally applied to the sides. A Manning-Stricklerdrag with n=0.02sm−1/3 is applied at the bottom bound-ary. The gravitational acceleration is set to 9.81ms−2 andthe fluid is initially at rest. Time discretisation is per-formed with Crank-Nicholson integration (i.e. θ= 1

2 ) anda time step of 600s. In this case the WD threshold is setat d0 = 0.5mm. The free surface is forced at the deepopen boundary with a sinusoidal variation of amplitude2m, such that water column thickness oscillates between3–7m, with a period of 12h.

In the series of tests considered here, the horizontal ex-tent is varied from 1.38 × 102m to 1.38 × 106m, centredabout the defined benchmark length of 1.38× 104m. Thisprovides a range of element aspect ratios from 10−4 to10−8, a domain aspect ratio up to 3.62× 10−6 and spatialscales spanning over 10 orders of magnitude in a single do-main. Element lengths are scaled with the domain length,such that element aspect ratio relative to global aspectratio is maintained, with the extrusion in the third direc-tion also scaled to preserve element shape. The time stepis also scaled to ensure the wave Courant number is con-stant. The WD threshold d0, and vertical extent are keptconstant across all cases.

The free surface evolution of the intermediate case witha horizontal extent of 1.38 × 104m is shown in figure 2

10

at 10min intervals, matching [22] and [61], for the initialdrying and then wetting phase, respectively. The resultsare physically reasonable and comparable to other formu-lations ([37] and [61] for example). In particular, the freesurface interface suffers from neither underestimation withnegative water column thickness, nor does it produce os-cillations during the wetting process observed in [22] forsome of the 10 methods examined. This behaviour is char-acteristic of the solutions across the range of aspect ratios.

Through a modification of the optimum aspect ratio pa-rameter a in (38), there is a corresponding change in theaspect ratio felt in the discrete pressure matrix of elementsin dry regions. The parameter a is varied over the rangea ∈ [10−4, 104] in a suite of 1001 simulations of the baseBalzano case. Solver iteration number is used as an indi-cator of conditioning, and plotted in figure 3 for both thepressure and velocity calculations as mean and maximumvalues over the course of a WD phase. The parameterrange has been spaced equally in log-space in order to givea good representation of the behaviour over the large rangeof domain aspect ratios. This is achieved with a discreteparameter space defined for a parameter p, such that

p ∈ 10((2s/(n−1)−1)m) : s ∈ Z, 0 ≤ s < n,for n, the number of distinct individual simulations span-ning the parameter space over m orders of magnitude ei-ther side of zero, such that p ∈ [10−m, 10m].

Whilst the conditioning of the velocity solver is largelyunaffected, the number of iterations required for pressureconvergence increases dramatically as the magnitude ofthe parameter a increases. As the aspect ratio parame-ter becomes acutely large with |a| → ∞, behaviour tendsto that of the system without the scheme applied. It isclear that the vertical relaxation scheme has a positiveimpact on conditioning, reducing the number of requirediterations in the pressure solution in this test by a factorof 20. With Picard iteration numbers also reduced as aconsequence, this effect is multiplied for significant overallperformance gains.

Changes in the parameter demonstrate that the schemesignificantly improves conditioning in the base case. Nowan optimal aspect ratio a = 1 is specified and actualchanges to the domain extents considered. Again a suiteof simulations are run to span the parameter space anddetermine conditioning, and the number of iterations re-quired for convergence of the pressure is shown in figure 4,with and without the relaxation conditioning. In the rangeconsidered, the improvement is reduced by a factor of upto 20 and results highlight that the approach eliminates adependence of conditioning on aspect ratio.

7.3. Second Balzano shelf channel benchmark

This case also originates in Balzano [22] and differs fromthe first by the inclusion of a shelf break in the slopedbathymetry, defined in Appendix C. The horizontal do-main is discretised in a way to ensure accurate bottomboundary representation, such that element faces align

with the discontinuous changes in surface gradient (fig-ure 5). Except for the change in bathymetry, discretisationproceeds in the same manner as the first Balzano case ofsection 7.2, and is again run over a range of aspect ratios.

The free surface evolution in the case with minimumelement aspect ratio 10−6 is shown in figure 5, again char-acteristic of the formulation over the range of aspect ratios.In addition to the oscillatory and retention problems al-ready mentioned, Balzano noticed a runoff problem withsome methods in this test case, where water remains onthe shelf during the dry period instead of flowing into thebasin. Like [37] and [61], the runoff is observed to be linearin time, the correct physical behaviour.

With the irregular bathymetry of this case, we considerthe effect of how the length scales that are passed to the re-laxation scheme are calculated, as discussed in section 4.7.The characteristic height ∆z varies both by element andover elements, and can be calculated at quadrature pointsfor increased accuracy. Noting the role of these lengthscales in the vertical velocity relaxation inverse time scale(38), we see that errors in how they are determined influ-ence conditioning in the same manner as that of perturba-tions of a from the optimum value of 1, except to a greaterdegree due to the quadratic dependence which, followingfigure 3, reduces the effectiveness of the conditioning.

Method Drying phase Wetting phase

max mean max mean

Minimum 512 475 512 475Maximum 305 303 281 269Mean 287 285 328 310Minimum capped 300 297 321 301Jacobian 310 281 264 259

Table 1: Pressure solver iterations to convergence in the secondBalzano benchmark with a domain aspect ratio of 3.62 × 10−6 forfive approaches to calculating characteristic height.

Five approaches are considered (table 1 and section 4.7).The methods ‘minimum’, ‘maximum’ and ‘mean’ each re-fer to the minimum, maximum and mean of the set of sixvertical lengths ∆z calculated from the four tetrahedralelement vertices. The minimum of these performs poorlyin all phases, so its value was limited by a lower bound inthe ‘minimum capped’ approach, which prevents the ap-plied absorption becoming too large. This produced bet-ter conditioning than the maximum in the drying phase,and whilst improved in the wetting phase, the maximumhere still produced better conditioning. The mean behavesvery well in the drying phase, but only satisfactorily dur-ing wetting. This implies all three of these norms are notcapturing all of the important parameters to determine anoptimum ∆z. The Jacobian approach using the determi-nant of a contracted transformation matrix at quadraturepoints provides the best conditioning during the wettingphase. The conditioning in the drying phase is not consis-tently the best, but the lowest mean number implies it is

11

1.0

0.5

0.0

-1.0

-1.5

-0.5

-2.01.40.40.0 0.8 1.00.6 1.20.2

1.0

0.5

0.0

-1.0

-1.5

-0.5

-2.01.40.40.0 0.8 1.00.6 1.20.2

Figure 5: The second Balzanochannel flow benchmark witha horizontal extent, in thispresented case, of 1.38 ×104m, corresponding to a min-imum element aspect ratio of∼ 10−6. The discretised hor-izontal surface (a) contains58 triangular elements witha characteristic length scale∼ 50 − 100m. Vertical sectionsshow the free surface positionat 10min intervals for the ini-tial drying phase (b) and dur-ing wetting (c).

best overall. In the Balzano shelf case examined here, thenumber of iterations required for convergence is approxi-mately halved by a careful consideration of the calculationof ∆z.

7.4. Thacker parabolic basin benchmark

The Thacker parabolic bowl [62] is an idealised oceanbasin that thins at its edges, with bathymetry defined inappendix D. It is a challenging free surface flow problemwith WD that has previously been used in intercomparisonstudies [20, 22, 37, 61]. An analytical solution for theevolution of the free surface is known (also in appendix D)when both dissipation and Coriolis are absent, and thecase suitable for the evaluation of spatial and temporalaccuracy, and volume conservation.

The base case domain size matches that of [20, 22, 37,61, 62] with a 880km horizontal extent, R = 430.62km,h0 =50m, η0 =2m, with a minimum water thickness of d0 =0.5m. No viscosity or drag terms result in a non-dampedfree surface oscillation with a 12h period. We make theassumption that in this domain the hydrostatic componentof the free surface perturbation dominates with the non-hydrostatic part small, and thus the solution converges tothe analytical function in appendix D.

Conditioning is examined for domain aspect ratios rang-ing over four orders of magnitude, from the base 5.68 ×10−5 down to 5.68×10−8. This is achieved through verticalscaling the domain and d0, with the maximum equilibriumwater column depth varying between 50m−5cm. With thecharacteristic horizontal edge length ∼ 104m close to theedges where the domain dries, element aspect ratios varysimilarly ∼ 5× 10−5 − 5 × 10−8. A cross section of theresulting single-layer basin domain for the h0 = 5cm caseis shown in figure 6(b), with the initial perturbation η0

ensuring a minimum thickness of d0 is applied.Edge element length scales are defined isotropically by

ε(r) = ∆x (9 |(R− |r − r0|)/R|+ 1) , (46)

which for the case ∆x=104m in figure 6, result in a rangefrom 100km in the middle down to 10km at a distance3.8× 105m from the centre, in an approach following [37].

Numerical evolution of the free surface for the highestaspect ratio case, shown in figure 7(a)-(b), is observed tofit the analytical solution very well, even with elements of a

-0.2-0.4 0.2 0.40.0 0.30.1-0.3 -0.1

1.0

0.0

-1.0

-2.0

-3.0

-4.0

-5.0

Figure 6: Thacker parabolic basin benchmark with a vertical ex-tent of 5cm. The discretised horizontal surface (a) follows the met-ric (46) for the case ∆x= 104m, containing 1,818 nodes and 3,750triangular elements and characteristic length scales ∼10 − 100km.Along the bisecting line, (b) shows the representation of the parabolicbathymetry together with equilibrium and initial perturbed free sur-face positions.

very high aspect ratio (5×10−8). Like the results from themore modest domain size a phase shift is observed, whichalso seen in [37], is a feature produced by the thin layer inthe dry areas. We can eliminate numerical dissipation in-herent in the scheme as a contributor, as we find that withsolves iterated to convergence, volume is conserved up to

12

1.5

1.0

0.0

-0.5

-1.0

-1.5

0.5

0.0

2.0

1.5

1.0

0.0

-0.5

-1.0

-2.0

-1.5

0.5

4.00.0 12.08.0 14.010.02.0 6.0

2.0

1.5

1.0

0.0

-0.5

-1.0

-1.5

0.5

Figure 7: Thacker parabolic benchmark, showing analytical (solid)and numerical (dashed) solutions. Evolution of η in the 5.68 × 10−8

aspect ratio domain at (a) the centre of the basin and (b) a distance424km from the centre marked * in figure 6(b). Radial velocity urevolution at the free free surface, in the base domain of figure 6, ata distance (c) 212km, and (d) 424km.

a relative factor of 1.0× 10−11, which is attributed to nu-merical round off error. This phase shift is reduced withan increase in mesh resolution [see 37], which contributesto the increase in accuracy observed in figure 9(b). In thetime series taken at the edge of the domain, it is clear when

-0.2-0.4 0.2 0.40.0 0.30.1-0.3 -0.1

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

Figure 8: Thacker benchmark analytical (solid) and numerical(dashed) η solutions along the vertical slice indicated in figure 6after thirty days, in a domain with global aspect ratio 5.68 × 10−8.

the location becomes dry in both the analytical and nu-merical solution, and where the factor of d0 is maintainedin the latter (here 0.5mm).

The radial velocity at the free surface at two locations ispresented in figure 7(c)-(d) at approximately the same rel-ative locations as those considered in [63], and is comparedto the analytical solution provided in appendix D. In themain body of fluid the solution is a very good match, withthe same shift observed in η as in figure 8 above and [37].Close to the edge of the basin, ur is not as well predictedas η. This is partly due to the continuous nature of thethin-film approach, which solves for u in both wet and dryregions. The spatial discretisation local to this point is rel-atively coarse, and additionally is not aligned to a radialdirection, which makes ur particularly challenging to cal-culate. This and the phase error, can be mitigated by in-creasing spatial resolution and constraining mesh structureto align with flow direction in inundation regions. Impor-tantly, accuracy of [37] is maintained, whilst the difficultyin solving the linear systems is much reduced.

The position of the free surface in a vertical slice of thedomain along the line indicated in figure 6 and after a pe-riod of thirty days, to include two each of the WD phases,is shown in figure 8. Spatially, the numerical solution isa good fit to the analytical solution and its resolution ofthe WD front comparable to studies in more modest as-pect ratio domains [37, 61]. The use of the vertical velocityrelaxation approach and iterative solvers for the linear sys-tems does not have a significant impact on accuracy of thesolution, and provides a formulation for high aspect ratiodomains that performs as well as those of modest size.

An evaluation of error e(t), at a time t, is made underan L2 norm of the absolute difference, such that

e(t) = ‖η(t)−max (ηa(t), h+ d0)‖2,Ω ,

=

(∫Ω

|η(t)−max (ηa(t), h+ d0)|2) 1

2

,

for h and ηa defined in appendix D. The minimum waterdepth is included in the analytical solution, since this isthe free surface height the formulation converges to, andthe domain Ω encompasses both wet and dry regions.

13

Figure 9: Thacker benchmark convergence proper-ties. Accuracy relative to (a) edge length in basedomain of figure 6 and with respect to aspect ra-tio in (b). Error is evaluated at the point of timethat the initial wetting period is complete. Linearand quadratic gradients are indicated by dashed lines.The diagonal cross, marked by *, points to a case witha relative tolerance reduced to 10−10.

Solution convergence with respect to the smallest hor-izontal characteristic edge length ∆x is considered in fig-ure 9(a) for the base domain, where the time step is lin-early scaled to maintain a constant CFL number. Mesheddomains are generated by scaling the metric (46). Withthis WD formulation we obtain the linear convergence inerror to characteristic edge length observed in [37].

The impact of domain aspect ratio on the accuracy ofthe calculation of free surface height after the initial wet-ting phase is considered in figure 9(b). Notably the er-ror does not increase significantly with an increase in themagnitude of the aspect ratio and is far from linear. Theincrease can be accounted for, to some extent, by the fixedrelative tolerance on the iterative solvers of the linear sys-tems. Adjusting this tolerance to increase convergencein cases with very small edge lengths could help to in-crease accuracy at this level. A small improvement in ac-curacy is seen in the highest aspect ratio case consideredin figure 9(b) where the relative error tolerance of 10−7

described in section 7 is reduced to 10−10. It is a signif-icant result that a solution can be found for these caseswith very high aspect ratios and additionally, that the ap-proach does not have an appreciable impact on accuracy.

7.5. Basin inundation

This case considers the inundation of water into an ini-tially dry basin, with the effect of bathymetric features onWD front propagation also examined. The base domain isshown in figure 10 and consists of a basin with horizontalextent 100m× 100m, and an inlet of width 10m, its centrepositioned 15m in from one of the corners. The domain isdiscretised with elements of a characteristic edge length of5m. The problem is forced with a normal inlet velocity of0.5ms−1 to model a levee breach into a flood plain on anurban scale.

To provide a more natural forcing, instead of applyinga flux directly on the boundary, the inlet is extended back10m and is maintained wet throughout by developing asloped bathymetry back, down to a depth of b= 20m, asseen figure 10(b). The normal inlet velocity is then appliedto the face that has been extended back, with velocity slipconditions on the adjacent sides. This was found to avoidproblems with the inflow at the edges of the breach. At the

0

Figure 10: Flood plain basin benchmark (a) horizontal domain. Cir-cular contours mark 1

10h0, 1

2h0 and 9

10h0, of the applied bathymetric

features. Solid and dashed lines mark the bathymetry cross-sectionsappearing in (b), where plain, hollow and hill case profiles are shown.

outflow on the far boundary at y= 100m, a natural Neu-mann condition is applied perpendicular to the boundary,such that ∂v/∂y = 0, for velocity v in the y−direction.All other boundaries are closed, with no normal flow con-ditions applied. Other velocity components are free andleft unconstrained.

To ensure accuracy of the calculation of prognostic vari-ables is not affected by the use of relatively large timesteps with potential impact on the conditioning analysis,∆t is set conservatively at 10s to give a maximum Courantnumber of 1.

14

Figure 11: Impact of the optimal aspect ratio parameter on solverconditioning in the flood plain basin benchmark over a WD phase.1001 individual simulations shown.

Figure 12: Pressure solver iterations to convergence in the flood plainbasin benchmark with respect to domain global aspect ratio, in 1001individual simulations, with and without conditioning applied.

Figure 13: Impact of horizontal viscosity νL of (45) on solver condi-tioning in the flood plain basin with hill protrusion benchmark overa WD phase. 1001 individual simulations shown.

In a similar approach taken for the Balzano slope case,we consider the influence of the optimum aspect ratio pa-rameter a on conditioning in the base domain with as-

pect ratio 104, over a parameter space spanned by 1001simulations shown in figure 11. Conditioning of the pres-sure solver is significantly improved, by over a factor of sixin this modest aspect ratio case. Again velocity is onlyslightly affected, and felt through the coupling, a con-sequence of better pressure conditioning. When varyingsimulation domain extent, with an optimal choice of a=1,similar behaviour is observed and shown in figure 12.

In practice large gradients in bathymetry have an im-pact on conditioning. This is studied with the introduc-tion of a depression in the domain to form a hollow andconversely, a raised hill. Both interact differently with theincoming wetting front. These features are introduced tothe domain with a Gaussian perturbation, which is definedat all points r on the horizontal surface of the domain byh(r) = h0e

− 12 ((r−r0)·σ)2, for h0 the maximum deviation in

height, which occurs at the centre where r= r0. The in-verse variance vector σ defines width, and consequentlythe gradient, of the obstacle. In the scales of the basecase, the magnitude of the perturbation |h0| is 5m, witha width of 10m, defined by σ =

(110 ,

110

). The perturba-

tion is positioned at 30m in from each of the boundingedges at the corner closest to the inlet. At the start, theminimum water thickness of d0 =1cm is applied above thebathymetry, to provide the initial thin dry flood basin.

The above is used to generate an inundation into a do-main containing a large hollow with h0 =−5m. Condition-ing is further decreased with the presence of the hollow,with a mean number of 105 iterations required in pres-sure for the modest aspect ratio case. Compared to theflat case, the number of iterations required increases at agreater rate, and the positive effect on conditioning of thevertical relaxation scheme is further pronounced. Addi-tionally, the large gradients in bathymetry adversely affectconditioning of the velocity solver early in the simulationwhere large velocities develop around the steep slopes tofill the hollow. This can be seen in the example snap-shot results shown in figure 14. Initially flow is strongfrom the breach, and predominantly flows into the hollow,whose surface oscillates in a similar manner to that seenin the Thacker parabolic bowl benchmark of section 7.4.Once the hollow is filled, the free surface peaks and a hy-draulic jump develops between the fast-flowing inlet fromthe breach and the formed lake. The fluid then gains mo-mentum in the direction of the inlet flow and is seen tobuild up on the opposite boundary. A clear front has de-veloped by this stage and begins to propagates across theplain towards the open boundary. It is also possible to seethe larger velocities that develop at the front and ahead inthe thin dry regions. This is motivation for the applicationof velocity conditioning discussed in the following.

In the case of the hill, with h0 =5m in the base domain,the effect on velocity is more significant, particularly asthe WD front meets the bathymetric intrusion. In thiscase it is necessary to apply a regularisation to the momen-tum equation to improve conditioning, which is achievedthrough an application of a bulk volume viscosity, as intro-

15

Figure 14: Inundation into a flood basin of side length 100m and threshold value 1cm containing a hollow bathymetric feature. Threesuccessive visualisations with (a) the hollow filling, (b) a hydraulic jump and (c) propagation further into the plain, are shown at 7350, 20400and 37650s into the simulation, respectively. The left panels contain contour plots of free surface perturbation, overlaid with magnitude-scaledvectors of depth-integrated velocity. The right panels present the 3D domain stretched in the vertical by a factor of 40, to better show thechange in free surface height, with the inlet breach and connecting reservoir seen on the right side. Contours of the magnitude of surfacevelocity are plotted together with vectors indicating the surface flow direction. Note the velocity fields presented are those in a continuousP1 space, calculated through a Galerkin projection from the discontinuous PDG

1 prognostic velocity field [see 49].

16

duced in section 6.2. The domain-wide horizontal viscosityνL, is varied over the range νL ∈ [10−4, 104] m2s−1 through(45) in the mid aspect ratio 10−6 case, with conditioningshown in figure 13. As a general trend, the number of it-erations required increases with strengthening of the hori-zontal viscosity. There is however a point at which there isa noticeable dip, where the increase in intensity improvesconditioning. This decrease in the mean number of itera-tions is due to improvement of conditioning made when thefront approaches and traverses the hill protrusion. Limit-ing application of this conditioning to dry regions and itsproximity, as described in section 6.2, allows WD fronts toencounter steep changes in bathymetry without the corre-sponding impact on conditioning of the velocity solver, inthis implicit and continuous WD formulation.

Increasing the bottom drag through the Manning-Strickler parameterisation in this region as outlined in sec-tion 6.1 also acts to improve conditioning. In particularlyhigh aspect ratio cases with steep bathymetry, the velocitysolver is too badly conditioned for efficient solution with aSSOR-GMRES iterative process without approaches suchas the horizontal viscosity and drag discussed.

8. Conclusion

In this paper a novel approach to efficiently modellingWD inundation processes in 3D, capturing non-hydrostaticand baroclinic physics, in the high aspect ratio domainsthat characterise geophysical systems has been proposed.

This has identified the ill-conditioning present in im-plicit continuum WD methods applied in fully 3D fluidflow models. Following a quantification of the highly spa-tial and temporally variable contributing factors, regular-isation of the governing weak form leads to a linear sys-tem that appears as a unit aspect ratio problem. Theresult is that the approach can be used to model WD inmultiscale geophysical domains, seamlessly alongside otherchallenging physics, such as baroclinic and non-hydrostaticflow, without a severe and limiting impact on the iterativesolvers typically required for efficient simulation of multi-physics 3D dynamics.

The approach has been demonstrated effective over awide range of spatial scales and correspondingly, aspectratios. The predicted behaviour on convergence is verifiedin numerical tests in both domain and element aspect ra-tios representing up to 8 orders of magnitude difference,with discrete domains containing spatial scales spanning10 orders of magnitude.

The approach imposes no restrictions on space andtime discretisation, permitting an arbitrarily flexible meshchoice (including generalised vertical coordinates), orderof representation and implicit time integration. All areimportant for system models simulating over a range ofscales and physics. Discretisation can be chosen largelyindependent of WD considerations, with for example, spa-tial resolution focused on local physics modelling demands.

Use of a combined p(p, η) variable strictly enforces con-sistency between the full 3D pressure and free surface per-

turbation. Notably there is no need to interpolate η andits derivatives from Γη to the internal domain Ω for in-clusion in the momentum calculation. Consistency withother fields and conservation are achieved by the overallFE approach, which can provide a high order continuumrepresentation. P1 − P1 and the heterogeneous elementpairing PDG

1 −P2 have been applied in the numerical tests.The implicit treatment of p(p, η) is inherited by p and η,and as a result, ∆t may be based solely on accuracy con-siderations and not stability when considering free surfacewave propagation. As discussed in [7] this may need care-ful consideration when a system is under-resolved with arelatively irregular bottom topography containing sharpgradients, or in high Froude number rapidly varying flow.

A limitation to note is that the free surface interface can-not become unduly complicated, including folds, since thefunction η is by definition injective with only a single pointpermitted to lie on the surface for any point within the do-main. As such it is not possible to model breaking waves, acommon limitation to all of the Eulerian approaches cited.

Unlike schemes applying additional viscosity or bed fric-tion based on empirical numerical measures that poten-tially lead to stabilisation through unphysical means, theapproach ensures physical consistency such that resultantsolutions are enforced to exist in the space of solutionsavailable to the original physically based weak form of thecontinuum governing equations (1)–(2). Physical consis-tency is verified in the numerical tests. Lastly, since theterms introduced specifically to improve conditioning areformulated in the continuum primitive form, this part ofthe approach could equally be applied in other WD imple-mentations for an arbitrary underlying discretisation.

This approach will not be optimum for some WD prob-lems, particularly due to the computational cost even withthe aspect ratio problem solved, where a single layer SWEapproximation is sufficient, or computational efficiencymay demand lower order methods for real-time tsunamiprediction, for example. However this approach now en-ables the modelling of physical phenomena not possiblepreviously, particularly those at the interfaces of tradi-tionally separate fields. With rapid ongoing developmentof computational resources, this approach and similar willgrow in use and become more common practice – a wayto bring WD to seamless massive multiscale multi-physicsEarth system models.

Acknowledgements

I would like to thank Christopher Pain for the helpful dis-cussions at all stages of this research, as well as his re-views of the manuscript and his encouragement to pub-lish. Additionally I am grateful for Matthew Piggott’sfeedback and support for this work. I am grateful to thethree anonymous reviewers for their valuable commentsand suggestions which helped improve the manuscript.I would also like to acknowledge support from FangxinFang who helped secure the European Commission Frame-work Programme 7 PEARL grant (Ref 603663), which

17

partly funded this work. Further support came fromthe UK Natural Environment Research Council (grantNE/G018391/1) and Engineering and Physical SciencesResearch Council (grant EP/I00405X/1). Computationalresources were provided by Imperial College and UK aca-demic HECToR/ARCHER HPC services.

A. Kinematic condition and nomenclature

Evolution of the free surface accommodating surface wavesrequires a further prognostic variable defining its heightη : Ω 7→ R (see figure 1) with the interface parametrisedby z = η(x, y), where η = 0 when the fluid is at rest andin equilibrium. Without loss of generality, the referenceframe is rotated to align z to the local gravitational di-rection, with ng = (0, 0, 1)T . An additional constraint isrequired and the assumption made that a fluid parcel onthe free surface remains there throughout time [64], whichwith the coordinates of a fluid parcel (x(t), y(t), z(t))T , iswritten η(x, y, z, t)=z, for t ∈ [0, T ), with time derivative

∂η

∂t= −∂η

∂x

∂x

∂t− ∂η

∂y

∂y

∂t+∂z

∂t= n · u,

for the surface normal vector n= (−∂η/∂x,−∂η/∂y, 1)T

and normalised form n=n/ |n|. Scaling by |n| and notingn · ng=1, gives the kinematic condition

n · ng∂η

∂t= n · u on Γη.

B. Finite element basis definitions

The weak form [65] of the governing equations is obtainedby an inner product with all test basis functions from aSobolev space S := H1(Ω) defined over the domain Ω withgeneralised first derivatives and an L2 inner product. TheGalerkin FE spatially discretised equations are found bylimiting S to a discrete subspace Sh ⊂ S, itself definedover a discrete representation of the domain, containing afinite number of spanning orthogonal trial functions. Theprognostic variables are represented

u :=∑

uiφi, p :=∑

pjψj , for ui, pi ∈ R,

and trial functions φi : Ω 7→ R3 and ψj : Ω 7→ R, and sumover the entire Sobolev spaces. Applications in this pa-per operate on meshes consisting of tetrahedral elements;with discontinuous piecewise linear functions and continu-ous piecewise quadratic functions for velocity and pressurerespectively, referred to as PDG

1 −P2 and introduced in [56].

C. Second Balzano benchmark bathymetry

h(x) =

x/2760, for x ∈ [0.0, 3.6] km,30/23, for x ∈ (3.6, 4.8] km,x/1380− 50/23, for x ∈ (4.8, 6.0) km,x/2760, for x ∈ [6.0, 13.8] km.

D. Thacker parabolic basin benchmark functions

Basin bathymetry is a parabola of the form

h(r) = h0(R2 − |r − r0|2)/R2,

for position vector r on the 2D horizontal surface, r0 locat-ing the disc centre, R the basin radius at rest, and h0 the

equilibrium water column depth at r = r0. The analyticalfree surface evolution, inferred from [62], is

ηa(r, t) = h0

( √1−η2

1−η cosωt− |r−r0|2

R2

(1− η2

(1−η cosωt)2−1

)−1

),

with η =(h0 + η0)2 − h2

0

(h0 + η0)2 + h20

, and ω2 =8gh0

R2,

where η is the initial free surface perturbation at r = r0,such that ηa(r0, 0) = η. Analytical horizontal velocitiesare calculated in [62] (with polar versions in [63]) and forthe examined cases reduce to

ur(r) =ωη |r−r0| sinωt

2(1− η cosωt), for ηa(r, t) > 0.

Model availabilityThe approach is implemented in the general purpose,arbitrarily unstructured, FE geophysics model Fluid-ity [38], https://fluidity-project.org, which is opensource, available under LGPL at https://github.com/

FluidityProject/fluidity, with verification tests spe-cific to the approach for high aspect ratio domains de-scribed.

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